OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..5000
FORMULA
a(n) = Sum_y A000005(k), where the sum is over all integer partitions of n and k is the number of parts.
From Seiichi Manyama, Jan 21 2022: (Start)
G.f.: 1 + Sum_{k>=1} A000005(k) * x^k/Product_{j=1..k} (1-x^j).
G.f.: 1 + Sum_{i>=1} Sum_{j>=1} x^(i*j)/Product_{k=1..i*j} (1-x^k). (End)
a(n) = Sum_{i=1..n} Sum_{j=1..n} A008284(n,i*j). - Ridouane Oudra, Apr 13 2023
EXAMPLE
The a(5) = 14 split partitions:
[5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
.
[4] [3] [2 1]
[1] [2] [1 1]
.
[3] [2]
[1] [2]
[1] [1]
.
[2]
[1]
[1]
[1]
.
[1]
[1]
[1]
[1]
[1]
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0 or i=1, numtheory
[tau](t+n), b(n, i-1, t)+b(n-i, min(n-i, i), t+1))
end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..50); # Alois P. Heinz, Jan 15 2019
MATHEMATICA
Table[Sum[Length[Divisors[Length[ptn]]], {ptn, IntegerPartitions[n]}], {n, 30}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1,
DivisorSigma[0, t+n], b[n, i-1, t] + b[n-i, Min[n-i, i], t+1]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 50] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)
PROG
(PARI) my(N=66, x='x+O('x^N)); Vec(1+sum(k=1, N, numdiv(k)*x^k/prod(j=1, k, 1-x^j))) \\ Seiichi Manyama, Jan 21 2022
(PARI) my(N=66, x='x+O('x^N)); Vec(1+sum(i=1, N, sum(j=1, N\i, x^(i*j)/prod(k=1, i*j, 1-x^k)))) \\ Seiichi Manyama, Jan 21 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 15 2019
STATUS
approved